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The Bermuda Triangle:
Weather, Electricity and Insurance
Derivatives
by
Hlyette Geman
University Paris IX Dauphine and ESSEC
September 1999
Introduction
Deregulation of electricity markets, well under way in the United States, has created for the U.S utilities and gas producers an environment of severe competition and will probably put European utilities under the same pressure once the Directives on gas and electricity deregulation become effective over the continent. The variability of revenues due to weather conditions that utilities have faced for a long time is now augmented by the effect on earnings of new entrants into the power market.
A well-known, and important result in the economic theory of insurance establishes that, under reasonable assumptions of risk aversion, an economic agent exposed to two sources of risk, one hedgeable, the other unhedgeable, will choose a higher coverage on the first risk than he would have if the latter one did not exist. An illustration of this result is provided by the development of the weather derivatives market in the U.S over the last two years, coinciding with the emergence of new shocks affecting the revenues of utilities.
The Chicago Mercantile Exchange is about to introduce standardized weather futures and options which will be traded on Globex, the exchange electronic platform. So far, weather derivatives have been sold by leading energy companies and insurance companies, mostly through the intermediation of a broker since these options are still OTC contracts. Another type of weather-related instruments, the so-called catastrophe options, were launched by the Chicago Board of Trade as early as December 1993. However these derivatives require, like most insurance products, a demonstration of loss and an evidence of the linkage of this loss to one of the nine well-defined catastrophic events (e.g., earthquake, hail, tornado) triggering the PCS (Property Claims Service) option index increments. Weather derivatives, in contrast, require no evidence of this type since the option payout is expressed in terms of a meteorological index. These bilaterally traded contracts can be combined in such a way that the risk exposure is reduced in accordance to the companys attitude toward risk.
Description of the weather contracts
Many forms of weather options such as precipitation, snowfall or windspeed options are available, covering a one-year or multiyear period. But the biggest volume so far has been observed on degree-days options which are related to daily average temperatures. More precisely, cooling and heating degree-days are defined as follows :
Daily CDD = max (daily average temperature 65 Fahrenheit, 0)
Daily HDD = max (65 Fahrenheit daily average temperature, 0)
and are meant to represent the deviations from a benchmark temperature of 65 Fahrenheit. Classically, a CDD (or Summer) season includes months from May to September, and HDD season months from November to March.
Moreover, in order to represent the magnitude of the season demand for electricity dedicated to air conditioner cooling (respectively for heating gas), the aggregation effect is reflected by the following payout of the CDD option at maturity
CDD (T) = Nominal Amount . Max INCORPORER Equation (1)
where n denotes the number of days in the exposure period as specified in the contract, I(t) is the average daily temperature registered at date t in the specified location and k is the strike price of the option expressed in degrees Fahrenheit. Hence, a cooling degree-day derivative is nothing but an Asian call option written on a daily CDD as the underlying source of risk.
In the same manner, the payout of a heating degree-day option at maturity is
HDD (T) = Nominal Amount . Max INCORPORER Equation (1)
i.e., the payout of an Asian put option written on heating-degree days.
We can notice that for both CDD and HDD derivatives, the option value is highly non linear with respect to the temperature index.
In the case of the option contracts introduced by the Chicago Mercantile Exchange in September 1999, the nominal amount is $ 100, a relatively small number meant to create liquidity (the number of degree-days during the months of January or July may be of the order of 1,000). The final settlement price is defined by the HDD or CDD (cumulative) index of the contract month as calculated by Earth Sat, and at this point the contracts exist for eight cities in the United States : Atlanta, Chicago, Cincinnati, Dallas, New-York, Philadelphia, Portland and Tucson.
In the case of the insurance derivatives, the Chicago Board of Trade has been using since December 1993 the services of independent statistical firms dedicated to insurance data Insurance Statistical Offices (ISO) in a first stage, Property Claim Services (PCS) since September 1995 for them to provide the final (and also intermediary) values of the loss catastrophic indexes associated with the nine regional derivatives contracts. In the same manner, Earth Satellite Corporation (Earth Sat), an international service firm, has been designated by the Chicago Mercantile Exchange to define the degree-day indexes. Earth Sat has developed remote sensing and geographic information technologies and the data it has provided over time have proven very accurate when compared with the data of the US National Climatic Data Center. The Globex electronic trading platform will not only allow transactions over 24 hours but also provide a price transparency particularly important for small investors who do not have a Meteorology Department within their firms.
The degree-days Futures contracts of the Chicago Mercantile Exchange trade, like the degree-days options, for each calendar month. They share this feature with all electricity Futures contracts traded in the United States or in Europe, property which has the merit of nullifying the calendar basis risk when hedging weather derivatives with electricity derivatives. The terminal value F(T) of a Future contract at maturity is defined as
F(T) = $100 ( INCORPORER Equation degree-days measured on day j by Earth Sat(
Hence, F(T) will be very high if the weather conditions have been extreme during the month of analysis. And economic actors whose revenues are hurt by these extreme temperatures will hedge their risk by buying at a date t prior to maturity (an appropriate number of) Futures contracts at the price F(t), hence cashing at maturity T the amount F(T) F(t) (positive if weather conditions have been more extreme than anticipated) which will offset their operating losses.
Several other observations are in order at this point :
For a utility or a gas producer, balance sheet can be managed using classical derivative contracts written on underlying equity, interest rates or exchange rates. But these instruments, as useful as they may be, do not provide protection against the business risk represented by volumetric risk, i.e., the uncertainty in revenues related to changes in demand for gas and power because of weather patterns. In a situation where inventories and storage cannot be an option because of the nature of the underlying commodity, weather derivatives (or volumetric energy derivatives, as we will discuss further on) can be structured to smooth the cash flow profile. Market players are generally reluctant to publicize transactions, because this would in particular indicate to competitors what risks they view as particularly dangerous for their earnings.
The fact that nearly all degree-day contracts are tied to the National Weather Service data guarantees the absence of opacity and almost excludes possible manipulations of the index, crucial properties to the eyes of users who may be legitimately concerned about the possibility of being taken advantage of by a sophisticated counterparty. The contract sites are US government sites and the final settlement of the option occurs when the weather Service publishes the official record, generally a few weeks later. Moreover, the US National Climatic Data Center maintains the American archive of weather data (where historical time series can be obtained) and also provides access to the World Meteorological Organization.
The three communities carefully watching the growth of the market of weather derivatives in the US are in agreement with the pricing and hedging arguments which will be made in this paper power marketers, power producers and insurance/reinsurance companies. The agricultural community is another obvious potential player but has not really been part of the transactions yet. More generally, it is estimated that 80% of all businesses are exposed to weather risk (construction companies, retail business, tourism industry and so forth) and that about $ 1 trillion of the $ 7 trillion US economy is weather-sensitive.
In order to create a liquid market through the existence of potential buyers and sellers, the derivative contracts are often capped, i.e., the option payout is defined as the right-hand side of (1) as long as it is no greater than a fixed threshold U, hence limiting the option sellers exposure to this number.
Keeping in mind the algebraic gain profile at maturity of an option buyer (respectively seller), the gain profile of a capped call option is the following at maturity
We can observe that a capped call option is nothing but a call spread (i.e., the combination of a long and short call options with different strikes). It was emphasized in Geman (1994) that call spreads, the most popular instruments among insurance derivatives, had the property of having a gain profile identical to the purchase of an excess of loss reinsurance contract, hence could be valuated by incorporating the insurance risk premium embedded in these XS of losses contracts into the option price.. In turn, it is not surprising to find the most sophisticated (re)insurance companies in the forefront of weather derivatives transactions.
Pricing CDD and HDD options
This problem is probably one of the hardest problems remaining to be solved in option pricing, the (non exhaustive) list of difficulties to address being the following.
The options have an Asian-type payout, leading to more mathematical complexities than classical options. These difficulties arise from the fact, that in the fundamental reference model of a log-normal distribution for the underlying state variable S
INCORPORER Equation (2)
(m and s constants), the dynamics described by (2) do not extend to a sum nor an arithmetic average. Hence, the Black-Scholes formula definitely does not apply and other answers need to be found. Geman-Yor (1993) use Bessel processes (which have the merit of being stable by additivity and of being related to the geometric Brownian motion through a time change) to obtain an exact analytical expression of the Laplace transform in time of the option price. The greeks are obtained with the same accuracy thanks to the linearity of the operators derivation and Laplace transform.
Another possible way of pricing Asian options is to use Monte-Carlo simulations. A single simulation of the index value over the whole period (0,T) provides one set of simulated daily values which, incorporated in formula (1), give in turn one realization of the payout of the CDD derivative. The average of these realizations over a number of Monte-Carlo simulations gives an approximation of the option price. Geman-Eydeland (1995) show that because of the smoothness of the Asian payout, a good approximation is obtained by a relatively low number of runs (e.g. , 10,000) ; but the same accuracy for the greeks (delta, gamma, vega) by necessity requires a higher number of simulations.
The evolution over the lifetime [0,T] of the option of the underlying source of risk I(t) needs to be properly modeled, taking into account seasonability effects over the year, stationarity of some season patterns across a period of several years and possible changes of the parameters over time. The local warming in a city or an airport site may be the result of global warming, or the product of expanding urbanization or may even be part of some secular climate cycle. Given the importance of an accurate modeling of the state variable in the option price, one has not only to analyze the temperature data series available from the National Climatic Data Center but also to use a survey of alterations in the built environment. This information has recently been systematically collected by insurance and reinsurance companies, sometimes with the help of specialized software companies. In the US, government agencies represent a very rich source of information (see for instance Figures 1 and 2) ; in fact, all the graphs in this chapter come from publicly available data.
Let us model the temperature index as an extension of the geometric Brownian motion. Representing the randomness of the world economy by the probability space ((, Ft, P),
where ( denotes the set of states of nature, Ft the filtration of information available at time t and P the objective probability measure, we model the dynamics of the average daily temperature I(t) by the stochastic differential equation
INCORPORER Equation (3)
where
the drift ((t, I(t)) may be mean-reverting to capture seasonal cyclical patterns, with a level of mean-reversion possibly varying with time to translate the global warming trend
the volatility ( should not be constant, since there seems to be a consensus on the greater volatility over time of the temperature, for a number of natural or man-made factors. However, it may be viewed admissible to take for ( either a deterministic function of time ((t) or to choose ( stochastic but depending only of the current level of the temperature index, i.e., a function ((t , I(t)). In both cases, there is no other source of randomness than the Brownian motion W(t), which will avoid incompleteness of the weather market and non unicity of the option price.
We know that the next step in the Black-Scholes-Merton proof is the construction at date t of a portfolio ! to be held up to date (t+dt) and comprising one call INCORPORER Equation This portfolio being riskless over the period [t, t+dt] has to provide, by no arbitrage arguments, a return equal to the riskfree rate r. This leads to the well-known partial differential equation satisfied by the call price
INCORPORER Equation (4)
with the boundary condition C(T)= max (0,S(T)-k)
In the case of options on interest rates such as caps, floors or bond options-which raise, at various levels, more difficulties than equity options - the introduction of a portfolio ! is still feasible, using in all cases bonds as substitutes for interest rates. In the case of weather derivatives, the same argument cannot be extended since weather is not a traded asset and there is no basic security such as stock or a Treasury bond whose price is uniquely related to a temperature index.
The Feynman-Kac theorem establishes that there exists a probability measure Q defined on ((, Ft)-where the probability space ( describes the set of states of nature and Ft the filtration of information available at date t - such that the solution to the partial differential equation (3) with its boundary condition can be written as
INCORPORER Equation (5)
This probability measure Q, called risk-adjusted, allows to price an option as the expectation of its terminal payout. It is obviously not equal to the statistical probability measure P under which data are collected and its identification is generally not straightforward (its unicity is insured by market completeness, obtained when the number of sources of randomness is no larger than the number of basic risky securities traded in the economy).
Some authors have proposed to compute the weather derivative price as the expectation (i.e., the sum of possible payouts weighted by their probabilities of occurrence) of its terminal value properly discounted and, in this order, to resort for instance to the Monte-Carlo simulations discussed earlier. This expectation is computed under the statistical probability measure P (meaning that the weights mentioned above do not reflect any correction for risk aversion), as if no risk premium was involved. For instance, Cao and Wei (1999) use a Lucas equilibrium framework approach to conclude that a zero market price of risk should be associated with weather derivatives values, hence justifying the use of the riskfree rate to derive these values as many practitioners do in the industry. This assertion is questionable since weather conditions do affect the aggregated economy in a significant way. To take an extreme example, consider the recent heat wave in Chicago (Summer 1999) during which 130 people died. Among the many losses attached to this event, there is a loss in human capital which is clearly not offset in an economic sense ; and the assumption of weather risk-neutrality of a so-called representative agent is not fully credible. As far as the weather market players are concerned, their view on the matter is clearly expressed by the large bid-ask spreads observed until lately on derivative prices (sometimes 100 % of the bid price).
The difficulty of identifying a (unique) probability measure Q, like the impossibility of delta hedging a weather derivative or the non-existence of a hedging portfolio comprising the weather derivative and other securities and totally riskless are different expressions of the same issue (which also affects credit derivatives for instance), namely the incompleteness of the weather derivative market. This is probably one of the most important problems yet to be solved in the financial theory of derivatives ( a well-known source of incompleteness which holds for all derivative markets today and is a legitimate subject of concern to practitioners is stochastic volatility).
Hedging a short position in weather derivatives
The existence of a perfect hedge for a given derivative - as in the Black-Scholes-Merton (1973) model - has the merit of providing all answers at once : the price of the option, equal to the cost of the hedging portfolio and obviously the hedging strategy itself. When this exact hedge does not exist, several types of answers can be proposed.
The introduction of a utility function for the representative agent, which leads to identify the derivative price as the solution of an optimization problem. The shortfalls of this approach reside in the questionable identification of this utility function and assumption of the same utility for all market players
The search of a so-called superhedging strategy H for the weather derivative, i.e., of a dynamically adjusted portfolio H such that at maturity
-C(T)+ H(T)g"0 in all states of the world
Unfortunately, the cost of this strategy is in most cases outside the bid-ask prevailing in the option market and nobody will buy the option at that price.
Carr-Geman-Madan (1999), observe the limited validity of approaches i) and ii) to price and hedge derivatives in incomplete markets and offer as an alternative the search for a portfolio H such that the position ( C + H) not be necessarily riskless but carry an acceptable risk
When C is a weather derivative, the hedge H will certainly comprise electricity contracts-either spot (such as financial assets ) or forwards and options. Since weather is the single most important external factor affecting the demand in power in the United States, one can try to represent this demand at date t for a future date tj (see Figures 3 ,4 and 5) as a function
w(t, tj) = f (I(tj), a1, a2, , an) (6)
where
I (tj) denotes the temperature which will be observed on day tj at the defined site
a1, a2, , an are parameters which may vary over time.
For instance, in the graphs (see Figure 6) reconstructed by Dischel (1999) using official US data, the three curves associated with regions as different Southeast Wisconsin and Southeast Washington have in common the property of showing a high use of residential electricity for low temperatures (below 45 Fahrenheit) and high temperatures (above 68 Fahrenheit) (the range of temperatures being obviously much narrower in Florida where it never gets very cold). Assuming that the quantity w(t, tj) can be represented as a second degree polynomial of I(tj) (not taking into account the other explanatory variables of electricity demand), one or the other root of this polynomial will allow, depending on the season, to express the temperature as a function of the demand.
Following Eydeland-Geman (1998), one may represent (see Figures 7 and 8) the Future price F(t, tj) as a function of the demand w(t, tj) and of the power stack function prevailing in the area of analysis. Using (5), the futures price becomes in turn a function of I(tj) and a hedge for the weather derivative can be elaborated using forward contracts. The right parts of the power stack functions 7 and 8 need to be analyzed in detail since they represent the extreme events, hence the large payoffs of weather derivatives
Another way to proceed is to analyze the power demand as a function of temperature through its deviation to the baseload and hedge the weather derivative using volumetric or swing electricity options which become the right protection when temperatures rise sharply. These instruments allow to call a bigger volume of power on a number of days, chosen by the option holder, during the lifetime of the option (with constraints on the total amount and possibly, the daily amount as well).
Lastly, to manage their risk exposure, hedgers of weather derivatives may try to benefit from the diversification effect created by several positions within the same region or across different regions, in the same manner as insurance companies hedge part of their underwriting risk through portfolios of insurance contracts. In both cases, the use of insurance derivatives may protect the catastrophic risk resulting from extreme weather events.
Conclusion
Weather specialists are developing a variety of products (precipitation-related instruments are the next ones under scruting, given their relevance for hydroelectricity in particular) enabling utilities to better manage weather risk, the largest source of financial uncertainty for many energy companies and a cause of revenue risk for a large number of sectors in the economy. These products create fresh ways for banks, (re)insurance companies and energy investors to take advantage of opportunities in this burgeoning market.
References :
Black Fisher and Myron Scholes (1973) The pricing of Options and Corporate Liabilities, Journal of Political Economy, 81.
Cao Melanie and Jason Wei (1999) Pricing Derivative Weather : an Equilibrium Approach,Working Paper
Carr Peter, Geman Hlyette and Dilip Madan (1999) Risk Management, Pricing and Hedging in Incomplete Markets , Working Paper.
Dischel Robert (1998) The Fledging Weather Market Takes Off, Applied Derivatives Trading, November.
Alexander Eydeland and Hlyette Geman (1998) Pricing Power derivatives, Risk, October.
Geman Hlyette (1994) Catastrophe Calls, Risk, September.
Geman Hlyette and Alexander Eydeland (1995) Domino Effect : Inverting the Laplace Transform, Risk, March
Geman Hlyette and Marc Yor (1993) Bessel Processes, Asia Options and Perpetuities, Mathematical Finance, 3.
Merton Robert (1973) Theory of Rational Option Pricing, Bell Journal Of Economics and Management Science, 4.
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